Least squares splines with free knots: global optimization approach

Splines with free knots have been extensively studied in regard to calculating the optimal knot positions. The dependence of the accuracy of approximation on the knot distribution is highly nonlinear, and optimisation techniques face a difficult problem of multiple local minima. The domain of the problem is a simplex, which adds to the complexity. We have applied a recently developed cutting angle method of deterministic global optimisation, which allows one to solve a wide class of optimisation problems on a simplex. The results of the cutting angle method are subsequently improved by local discrete gradient method. The resulting algorithm is sufficiently fast and guarantees that the global minimum has been reached. The results of numerical experiments are presented.

[1]  Diethard Pallaschke,et al.  Foundations of mathematical optimization : convex analysis without linearity , 1997 .

[2]  Lynn Margaret Batten,et al.  Fast Algorithm for the Cutting Angle Method of Global Optimization , 2002, J. Glob. Optim..

[3]  Yingkang Hu,et al.  An algorithm for data reduction using splines with free knots , 1993 .

[4]  Hubert Schwetlick,et al.  Least squares approximation by splines with free knots , 1995 .

[5]  H. Tuy.  Convex analysis and global optimization , 1998 .

[6]  Alexander Rubinov,et al.  Lipschitz programming via increasing convex-along-rays functions * , 1999 .

[7]  Reiner Horst,et al.  Introduction to Global Optimization (Nonconvex Optimization and Its Applications) , 2002 .

[8]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[9]  A. Rubinov.  Abstract Convexity and Global Optimization , 2000 .

[10]  Günther Nürnberger,et al.  Algorithms for piecewise polynomials and splines with free knots , 1989 .

[11]  B. M. Glover,et al.  Cutting angle methods in global optimization , 1999 .

[12]  H Lanshammar,et al.  On practical evaluation of differentiation techniques for human gait analysis. , 1982, Journal of biomechanics.

[13]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[14]  Hubert Schwetlick,et al.  Constrained approximation by splines with free knots , 1997 .

[15]  A. Wathen,et al.  On the Best Least Squares Approximation of Continuous Functions using Linear Splines with Free Knots , 1991 .

[16]  T. Lyche,et al.  A Data-Reduction Strategy for Splines with Applications to the Approximation of Functions and Data , 1988 .

[17]  Adil M. Bagirov,et al.  Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex , 2000, Ann. Oper. Res..

[18]  G. Beliakov.  Shape preserving approximation using least squares splines , 2000, Analysis in Theory and Applications.

[19]  A. M. Rubinov,et al.  MODIFIED VERSIONS OF THE CUTTING ANGLE METHOD , 2001 .

[20]  T. Lyche,et al.  The Sensitivity of a Spline Function to Perturbations of the Knots , 1999 .

[21]  Diethard Pallaschke,et al.  Foundations of Mathematical Optimization , 1997 .

[22]  Larry L. Schumaker,et al.  Shape-preserving knot removal , 1996, Comput. Aided Geom. Des..

[23]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[24]  Regina Hunter Mladineo.  An algorithm for finding the global maximum of a multimodal, multivariate function , 1986, Math. Program..

[25]  D. Jupp.  Approximation to Data by Splines with Free Knots , 1978 .

[26]  Paul Dierckx,et al.  Curve and surface fitting with splines , 1994, Monographs on numerical analysis.